Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  coe1tmmul2 Structured version   Visualization version   GIF version

Theorem coe1tmmul2 20915
 Description: Coefficient vector of a polynomial multiplied on the right by a term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Hypotheses
Ref Expression
coe1tm.z 0 = (0g𝑅)
coe1tm.k 𝐾 = (Base‘𝑅)
coe1tm.p 𝑃 = (Poly1𝑅)
coe1tm.x 𝑋 = (var1𝑅)
coe1tm.m · = ( ·𝑠𝑃)
coe1tm.n 𝑁 = (mulGrp‘𝑃)
coe1tm.e = (.g𝑁)
coe1tmmul.b 𝐵 = (Base‘𝑃)
coe1tmmul.t = (.r𝑃)
coe1tmmul.u × = (.r𝑅)
coe1tmmul.a (𝜑𝐴𝐵)
coe1tmmul.r (𝜑𝑅 ∈ Ring)
coe1tmmul.c (𝜑𝐶𝐾)
coe1tmmul.d (𝜑𝐷 ∈ ℕ0)
Assertion
Ref Expression
coe1tmmul2 (𝜑 → (coe1‘(𝐴 (𝐶 · (𝐷 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
Distinct variable groups:   𝑥, 0   𝑥,𝐶   𝑥,𝐷   𝑥,𝐾   𝑥,   𝑥,𝐴   𝑥,𝑁   𝑥,𝑃   𝑥,𝑋   𝜑,𝑥   𝑥,𝑅   𝑥, ·   𝑥, ×   𝑥,
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem coe1tmmul2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 coe1tmmul.r . . 3 (𝜑𝑅 ∈ Ring)
2 coe1tmmul.a . . 3 (𝜑𝐴𝐵)
3 coe1tmmul.c . . . 4 (𝜑𝐶𝐾)
4 coe1tmmul.d . . . 4 (𝜑𝐷 ∈ ℕ0)
5 coe1tm.k . . . . 5 𝐾 = (Base‘𝑅)
6 coe1tm.p . . . . 5 𝑃 = (Poly1𝑅)
7 coe1tm.x . . . . 5 𝑋 = (var1𝑅)
8 coe1tm.m . . . . 5 · = ( ·𝑠𝑃)
9 coe1tm.n . . . . 5 𝑁 = (mulGrp‘𝑃)
10 coe1tm.e . . . . 5 = (.g𝑁)
11 coe1tmmul.b . . . . 5 𝐵 = (Base‘𝑃)
125, 6, 7, 8, 9, 10, 11ply1tmcl 20911 . . . 4 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) ∈ 𝐵)
131, 3, 4, 12syl3anc 1368 . . 3 (𝜑 → (𝐶 · (𝐷 𝑋)) ∈ 𝐵)
14 coe1tmmul.t . . . 4 = (.r𝑃)
15 coe1tmmul.u . . . 4 × = (.r𝑅)
166, 14, 15, 11coe1mul 20909 . . 3 ((𝑅 ∈ Ring ∧ 𝐴𝐵 ∧ (𝐶 · (𝐷 𝑋)) ∈ 𝐵) → (coe1‘(𝐴 (𝐶 · (𝐷 𝑋)))) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))))))
171, 2, 13, 16syl3anc 1368 . 2 (𝜑 → (coe1‘(𝐴 (𝐶 · (𝐷 𝑋)))) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))))))
18 eqeq2 2810 . . . 4 ((((coe1𝐴)‘(𝑥𝐷)) × 𝐶) = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = (((coe1𝐴)‘(𝑥𝐷)) × 𝐶) ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
19 eqeq2 2810 . . . 4 ( 0 = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = 0 ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
20 coe1tm.z . . . . . . 7 0 = (0g𝑅)
211adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑅 ∈ Ring)
22 ringmnd 19304 . . . . . . . 8 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
2321, 22syl 17 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑅 ∈ Mnd)
24 ovex 7169 . . . . . . . 8 (0...𝑥) ∈ V
2524a1i 11 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (0...𝑥) ∈ V)
26 simprr 772 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷𝑥)
274adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷 ∈ ℕ0)
28 simprl 770 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑥 ∈ ℕ0)
29 nn0sub 11938 . . . . . . . . . 10 ((𝐷 ∈ ℕ0𝑥 ∈ ℕ0) → (𝐷𝑥 ↔ (𝑥𝐷) ∈ ℕ0))
3027, 28, 29syl2anc 587 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝐷𝑥 ↔ (𝑥𝐷) ∈ ℕ0))
3126, 30mpbid 235 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥𝐷) ∈ ℕ0)
3227nn0ge0d 11949 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 0 ≤ 𝐷)
33 nn0re 11897 . . . . . . . . . . 11 (𝑥 ∈ ℕ0𝑥 ∈ ℝ)
3433ad2antrl 727 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑥 ∈ ℝ)
354nn0red 11947 . . . . . . . . . . 11 (𝜑𝐷 ∈ ℝ)
3635adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷 ∈ ℝ)
3734, 36subge02d 11224 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (0 ≤ 𝐷 ↔ (𝑥𝐷) ≤ 𝑥))
3832, 37mpbid 235 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥𝐷) ≤ 𝑥)
39 fznn0 12997 . . . . . . . . 9 (𝑥 ∈ ℕ0 → ((𝑥𝐷) ∈ (0...𝑥) ↔ ((𝑥𝐷) ∈ ℕ0 ∧ (𝑥𝐷) ≤ 𝑥)))
4039ad2antrl 727 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((𝑥𝐷) ∈ (0...𝑥) ↔ ((𝑥𝐷) ∈ ℕ0 ∧ (𝑥𝐷) ≤ 𝑥)))
4131, 38, 40mpbir2and 712 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥𝐷) ∈ (0...𝑥))
421ad2antrr 725 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑅 ∈ Ring)
43 eqid 2798 . . . . . . . . . . . . 13 (coe1𝐴) = (coe1𝐴)
4443, 11, 6, 5coe1f 20850 . . . . . . . . . . . 12 (𝐴𝐵 → (coe1𝐴):ℕ0𝐾)
452, 44syl 17 . . . . . . . . . . 11 (𝜑 → (coe1𝐴):ℕ0𝐾)
4645ad2antrr 725 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (coe1𝐴):ℕ0𝐾)
47 elfznn0 12998 . . . . . . . . . . 11 (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℕ0)
4847adantl 485 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℕ0)
4946, 48ffvelrnd 6830 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1𝐴)‘𝑦) ∈ 𝐾)
50 eqid 2798 . . . . . . . . . . . . 13 (coe1‘(𝐶 · (𝐷 𝑋))) = (coe1‘(𝐶 · (𝐷 𝑋)))
5150, 11, 6, 5coe1f 20850 . . . . . . . . . . . 12 ((𝐶 · (𝐷 𝑋)) ∈ 𝐵 → (coe1‘(𝐶 · (𝐷 𝑋))):ℕ0𝐾)
5213, 51syl 17 . . . . . . . . . . 11 (𝜑 → (coe1‘(𝐶 · (𝐷 𝑋))):ℕ0𝐾)
5352ad2antrr 725 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (coe1‘(𝐶 · (𝐷 𝑋))):ℕ0𝐾)
54 fznn0sub 12937 . . . . . . . . . . 11 (𝑦 ∈ (0...𝑥) → (𝑥𝑦) ∈ ℕ0)
5554adantl 485 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑥𝑦) ∈ ℕ0)
5653, 55ffvelrnd 6830 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) ∈ 𝐾)
575, 15ringcl 19311 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ ((coe1𝐴)‘𝑦) ∈ 𝐾 ∧ ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) ∈ 𝐾) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) ∈ 𝐾)
5842, 49, 56, 57syl3anc 1368 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) ∈ 𝐾)
5958fmpttd 6857 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))):(0...𝑥)⟶𝐾)
601ad2antrr 725 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝑅 ∈ Ring)
613ad2antrr 725 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝐶𝐾)
624ad2antrr 725 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝐷 ∈ ℕ0)
63 eldifi 4054 . . . . . . . . . . . . 13 (𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)}) → 𝑦 ∈ (0...𝑥))
6463, 54syl 17 . . . . . . . . . . . 12 (𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)}) → (𝑥𝑦) ∈ ℕ0)
6564adantl 485 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (𝑥𝑦) ∈ ℕ0)
66 eldifsn 4680 . . . . . . . . . . . 12 (𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)}) ↔ (𝑦 ∈ (0...𝑥) ∧ 𝑦 ≠ (𝑥𝐷)))
67 simplrl 776 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑥 ∈ ℕ0)
6867nn0cnd 11948 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑥 ∈ ℂ)
6947nn0cnd 11948 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℂ)
7069adantl 485 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℂ)
7168, 70nncand 10994 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑥 − (𝑥𝑦)) = 𝑦)
7271eqcomd 2804 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 = (𝑥 − (𝑥𝑦)))
73 oveq2 7144 . . . . . . . . . . . . . . . 16 (𝐷 = (𝑥𝑦) → (𝑥𝐷) = (𝑥 − (𝑥𝑦)))
7473eqeq2d 2809 . . . . . . . . . . . . . . 15 (𝐷 = (𝑥𝑦) → (𝑦 = (𝑥𝐷) ↔ 𝑦 = (𝑥 − (𝑥𝑦))))
7572, 74syl5ibrcom 250 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝐷 = (𝑥𝑦) → 𝑦 = (𝑥𝐷)))
7675necon3d 3008 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑦 ≠ (𝑥𝐷) → 𝐷 ≠ (𝑥𝑦)))
7776impr 458 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ (𝑦 ∈ (0...𝑥) ∧ 𝑦 ≠ (𝑥𝐷))) → 𝐷 ≠ (𝑥𝑦))
7866, 77sylan2b 596 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → 𝐷 ≠ (𝑥𝑦))
7920, 5, 6, 7, 8, 9, 10, 60, 61, 62, 65, 78coe1tmfv2 20914 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) = 0 )
8079oveq2d 7152 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = (((coe1𝐴)‘𝑦) × 0 ))
815, 15, 20ringrz 19338 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ ((coe1𝐴)‘𝑦) ∈ 𝐾) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
8242, 49, 81syl2anc 587 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
8363, 82sylan2 595 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
8480, 83eqtrd 2833 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥𝐷)})) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = 0 )
8584, 25suppss2 7850 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))) supp 0 ) ⊆ {(𝑥𝐷)})
865, 20, 23, 25, 41, 59, 85gsumpt 19079 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = ((𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))‘(𝑥𝐷)))
87 fveq2 6646 . . . . . . . . 9 (𝑦 = (𝑥𝐷) → ((coe1𝐴)‘𝑦) = ((coe1𝐴)‘(𝑥𝐷)))
88 oveq2 7144 . . . . . . . . . 10 (𝑦 = (𝑥𝐷) → (𝑥𝑦) = (𝑥 − (𝑥𝐷)))
8988fveq2d 6650 . . . . . . . . 9 (𝑦 = (𝑥𝐷) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) = ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷))))
9087, 89oveq12d 7154 . . . . . . . 8 (𝑦 = (𝑥𝐷) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))))
91 eqid 2798 . . . . . . . 8 (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))
92 ovex 7169 . . . . . . . 8 (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))) ∈ V
9390, 91, 92fvmpt 6746 . . . . . . 7 ((𝑥𝐷) ∈ (0...𝑥) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))‘(𝑥𝐷)) = (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))))
9441, 93syl 17 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))‘(𝑥𝐷)) = (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))))
9528nn0cnd 11948 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝑥 ∈ ℂ)
9627nn0cnd 11948 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐷 ∈ ℂ)
9795, 96nncand 10994 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑥 − (𝑥𝐷)) = 𝐷)
9897fveq2d 6650 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷))) = ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷))
993adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → 𝐶𝐾)
10020, 5, 6, 7, 8, 9, 10coe1tmfv1 20913 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) = 𝐶)
10121, 99, 27, 100syl3anc 1368 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘𝐷) = 𝐶)
10298, 101eqtrd 2833 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷))) = 𝐶)
103102oveq2d 7152 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (((coe1𝐴)‘(𝑥𝐷)) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥 − (𝑥𝐷)))) = (((coe1𝐴)‘(𝑥𝐷)) × 𝐶))
10486, 94, 1033eqtrd 2837 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℕ0𝐷𝑥)) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = (((coe1𝐴)‘(𝑥𝐷)) × 𝐶))
105104anassrs 471 . . . 4 (((𝜑𝑥 ∈ ℕ0) ∧ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = (((coe1𝐴)‘(𝑥𝐷)) × 𝐶))
1061ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑅 ∈ Ring)
1073ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐶𝐾)
1084ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐷 ∈ ℕ0)
10954ad2antll 728 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) ∈ ℕ0)
11054nn0red 11947 . . . . . . . . . . . . 13 (𝑦 ∈ (0...𝑥) → (𝑥𝑦) ∈ ℝ)
111110ad2antll 728 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) ∈ ℝ)
11233ad2antlr 726 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑥 ∈ ℝ)
11335ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐷 ∈ ℝ)
11447ad2antll 728 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑦 ∈ ℕ0)
115114nn0ge0d 11949 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 0 ≤ 𝑦)
11647nn0red 11947 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℝ)
117116ad2antll 728 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑦 ∈ ℝ)
118112, 117subge02d 11224 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (0 ≤ 𝑦 ↔ (𝑥𝑦) ≤ 𝑥))
119115, 118mpbid 235 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) ≤ 𝑥)
120 simprl 770 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → ¬ 𝐷𝑥)
121112, 113ltnled 10779 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥 < 𝐷 ↔ ¬ 𝐷𝑥))
122120, 121mpbird 260 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝑥 < 𝐷)
123111, 112, 113, 119, 122lelttrd 10790 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (𝑥𝑦) < 𝐷)
124111, 123gtned 10767 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → 𝐷 ≠ (𝑥𝑦))
12520, 5, 6, 7, 8, 9, 10, 106, 107, 108, 109, 124coe1tmfv2 20914 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)) = 0 )
126125oveq2d 7152 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = (((coe1𝐴)‘𝑦) × 0 ))
12745ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (coe1𝐴):ℕ0𝐾)
128127, 114ffvelrnd 6830 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → ((coe1𝐴)‘𝑦) ∈ 𝐾)
129106, 128, 81syl2anc 587 . . . . . . . . 9 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (((coe1𝐴)‘𝑦) × 0 ) = 0 )
130126, 129eqtrd 2833 . . . . . . . 8 (((𝜑𝑥 ∈ ℕ0) ∧ (¬ 𝐷𝑥𝑦 ∈ (0...𝑥))) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = 0 )
131130anassrs 471 . . . . . . 7 ((((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))) = 0 )
132131mpteq2dva 5126 . . . . . 6 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ 0 ))
133132oveq2d 7152 . . . . 5 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )))
1341, 22syl 17 . . . . . . 7 (𝜑𝑅 ∈ Mnd)
13520gsumz 17995 . . . . . . 7 ((𝑅 ∈ Mnd ∧ (0...𝑥) ∈ V) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
136134, 24, 135sylancl 589 . . . . . 6 (𝜑 → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
137136ad2antrr 725 . . . . 5 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
138133, 137eqtrd 2833 . . . 4 (((𝜑𝑥 ∈ ℕ0) ∧ ¬ 𝐷𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = 0 )
13918, 19, 105, 138ifbothda 4462 . . 3 ((𝜑𝑥 ∈ ℕ0) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦))))) = if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 ))
140139mpteq2dva 5126 . 2 (𝜑 → (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 𝑋)))‘(𝑥𝑦)))))) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
14117, 140eqtrd 2833 1 (𝜑 → (coe1‘(𝐴 (𝐶 · (𝐷 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷𝑥, (((coe1𝐴)‘(𝑥𝐷)) × 𝐶), 0 )))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3441   ∖ cdif 3878  ifcif 4425  {csn 4525   class class class wbr 5031   ↦ cmpt 5111  ⟶wf 6321  ‘cfv 6325  (class class class)co 7136  ℂcc 10527  ℝcr 10528  0cc0 10529   < clt 10667   ≤ cle 10668   − cmin 10862  ℕ0cn0 11888  ...cfz 12888  Basecbs 16478  .rcmulr 16561   ·𝑠 cvsca 16564  0gc0g 16708   Σg cgsu 16709  Mndcmnd 17906  .gcmg 18220  mulGrpcmgp 19236  Ringcrg 19294  var1cv1 20815  Poly1cpl1 20816  coe1cco1 20817 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-iin 4885  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-se 5480  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-isom 6334  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-of 7391  df-ofr 7392  df-om 7564  df-1st 7674  df-2nd 7675  df-supp 7817  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-2o 8089  df-oadd 8092  df-er 8275  df-map 8394  df-pm 8395  df-ixp 8448  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-fsupp 8821  df-oi 8961  df-card 9355  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11629  df-2 11691  df-3 11692  df-4 11693  df-5 11694  df-6 11695  df-7 11696  df-8 11697  df-9 11698  df-n0 11889  df-z 11973  df-dec 12090  df-uz 12235  df-fz 12889  df-fzo 13032  df-seq 13368  df-hash 13690  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-sca 16576  df-vsca 16577  df-tset 16579  df-ple 16580  df-0g 16710  df-gsum 16711  df-mre 16852  df-mrc 16853  df-acs 16855  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-mhm 17951  df-submnd 17952  df-grp 18101  df-minusg 18102  df-sbg 18103  df-mulg 18221  df-subg 18272  df-ghm 18352  df-cntz 18443  df-cmn 18904  df-abl 18905  df-mgp 19237  df-ur 19249  df-ring 19296  df-subrg 19530  df-lmod 19633  df-lss 19701  df-psr 20600  df-mvr 20601  df-mpl 20602  df-opsr 20604  df-psr1 20819  df-vr1 20820  df-ply1 20821  df-coe1 20822 This theorem is referenced by:  coe1tmmul2fv  20917  coe1sclmul2  20923
 Copyright terms: Public domain W3C validator